I have encountered situations like this in the past: you are working with two different people, who could see each other as competitors, or as collaborators.
It is in your best interest to encourage the collaboration as much as possible. You are already halfway there - your friend has expressed interest in communication with the family-related solicitor. It is now up to you to help facilitate this communication.
A key to not sounding demanding is to try to frame the meeting as something that is in the best interest of the family-solicitor. You are coming to them from the assumption that of course they would want to talk to the second solicitor (perhaps to ensure that they are able to share information, avoid redundancies, and optimize their individual plans as much as possible).
Perhaps something along these lines:
"Hi [firstname], I don't know if you were aware, but I have a friend who is a solicitor who is helping me to handle the business side of the divorce. They thought it would be best if you two could have a conversation, to ensure that you can collaborate as best as possible and avoid working on potentially overlapping areas. Is that ok? I am happy to set up the call, or give you their contact information. Of course, your normal rate will also apply for any time spent working with them."
Also, it is important to be clear with the friend/business-solicitor in advance what are the parameters under which you are employing the family solicitor, and to hear from them how they best see things working out between the two professionals. This puts you in the best position to set up a meeting of minds between the two that can be friendly and productive.